Lower bounds on the essential dimension of reductive groups
Danny Ofek
TL;DR
The paper develops a general method to prove lower bounds on the essential dimension of split reductive groups by relating ed(G) to ed of centralizers of finite diagonalizable subgroups. It uses loop torsors and anisotropic torsors, together with a decomposition and descent framework for Henselian valued fields, to translate lower bounds into centralizer data, and then to symmetric-rank computations of Weyl-group actions on character lattices. This yields new concrete bounds for key groups, notably ed(E_8;2) and related bounds for E_8, E_7, E_6, HSpin16, and PGL-type groups, as well as a systematic, root-theoretic method for generating centralizers with favorable essential-dimension properties. The results have broad impact for understanding the parameter complexity of torsors and central simple algebras associated with these groups, and they sharpen long-standing bounds via a unifying, algebraic-geometry framework grounded in loop torsors and valuation theory.
Abstract
We introduce a technique for proving lower bounds on the essential dimension of split reductive groups. As an application, we strengthen the best previously known lower bounds for various split simple algebraic groups, most notably for the exceptional group $E_8$. In the case of the projective linear group $\operatorname{PGL}_n$, we recover A. Merkurjev's celebrated lower bound with a simplified proof. Our technique relies on decompositions of loop torsors over valued fields due to P. Gille and A. Pianzola.